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Contents
Chapter 1 Oscillation-Preserving Integrators For Highly Oscillatory Systems of Second-Order Odes 1
1.1 Introduction 1
1.2 Standard Runge-Kutta-Nystrom Schemes From The Matrix-Variation-Of-Constants Formula 5
1.3 Erkn Integrators And Arkn Methods Based On The Matrix-Variation-Of-Constants Formula 6
1.3.1 Arkn Integrators 7
1.3.2 Erkn Integrators 8
1.4 Oscillation-Preserving Integrators 11
1.5 Towards Highly Oscillatory Nonlinear Hamiltonian Systems 13
1.5.1 Ssmerkn Integrators 14
1.5.2 Trigonometric Fourier Collocation Methods 15
1.5.3 The Aavf Method And Avf Formula 18
1.6 Other Concerns Relating To Highly Oscillatory Problems 21
1.6.1 Gautschi-Iype Methods 21
1.6.2 General Erkn Methods For (1.1) 21
1.6.3 Towards The Application To Semilinear Kg Equations 22
1.7 Numerical Experiments 26
1.8 Conclusions And Discussion 36
References 37
Chapter 2 Continuous-Stage Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 42
2.1 Introduction 42
2.2 Extended Runge-Kutta-Nystrom Methods 45
2.3 Continuous-Stage Erkn Methods And Order Conditions 47
2.4 Energy-Preserving Conditions And Symmetric Conditions 50
2.5 Linear Stability Analysis 53
2.6 Construction of Cserkn Methods 55
2.6.1 The Case of Order Two 56
2.6.2 The Case of Order Four 57
2.7 Numerical Experiments 59
2.8 Conclusions And Discussions 63
References 64
Chapter 3 Stability And Convergence Analysis of Erkn Integrators For Second-Order Odes With Highly Oscillatory Solutions 68
3.1 Introduction 68
3.2 Nonlinear Stability And Convergence Analysis For Erkn Integrators 72
3.2.1 Nonlinear Stability of The Matrix-Yariation-Of-Constants Formula 72
3.2.2 Nonlinear Stability And Convergence of Erkn Integrators 77
3.3 Erkn Integrators With Fourier Pseudospectral Discretisation For Semilinear Wave Equations 83
3.3.1 Time Discretisation: Erkn Time Integrators 84
3.3.2 Spatial Discretisation: Fourier Pseudospectral Method 85
3.3.3 Error Bounds of The Erkn-Fp Method (3.57)-(3.58) 87
3.4 Numerical Experiments 97
3.5 Conclusions 107
References 107
Chapter 4 Functionally-Fitted Energy -Preserving Integrators For Poisson Systems 111
4.1 Introduction 111
4.2 Functionally-Fitted Ep Integrators 113
4.3 Implementation Issues 115
4.4 The Existence£¬ Uniqueness And Smoothness 117
4.5 Algebraic Order 120
4.6 Practical FFEP Integrators 123
4.7 Numerical Experiments 126
4.8 Conclusions 129
References 130
Chapter 5 Exponential Collocation Methods For Conservative Or Dissipative Systems 133
5.1 Introduction 133
5.2 Formulation of Methods 135
5.3 Methods For Second-Order Odes With Highly Oscillatory Solutions 138
5.4 Energy-Preserving Analysis 140
5.5 Existence£¬ Uniqueness And Smoothness of The Solution 142
5.6 Algebraic Order 144
5.7 Application In Stiff Gradient Systems 147
5.8 Practical Examples of Exponential Collocation Methods 148
5.8.1 An Example of Ecr Methods 148
5.8.2 An Example of Tcr Methods 148
5.8.3 An Example of Rkncr Methods 149
5.9 Numerical Experiments 150
5.10 Concluding Remarks And Discussions 156
References 157
Chapter 6 Volume-Preserving Exponential Integrators 161
6.1 Introduction 161
6.2 Exponential Integrators 163
6.3 Vp Condition of Exponential Integrators 164
6.4 Vp Results For Different Vector Fields 167
6.4.1 Vector Fields In 167
6.4.2 Vector Fields In 168
6.4.3 Vector Fields In 170
6.4.4 Vector Fields In (2) 171
6.5 Applications To Various Problems 173
6.5.1 Highly Oscillatory Second-Order Systems 173
6.5.2 Separable Partitioned Systems 176
6.5.3 Other Applications 178
6.6 Numerical Examples 179
6.7 Conclusions 188
References 188
Chapter 7 Global Error Bounds of One-Stage Explicit Erkn Integrators For Semilinear Wave Equations 191
7.1 Introduction 191
7.2 Preliminaries 192
7.2.1 Spectral Semidiscretisation In Space 192
7.2.2 Erkn Integrators 194
7.3 Main Result 195
7.4 The Lower-Order Error Bounds In Higher-Order Sobolev Spaces 196
7.4.1 Regularity Over One Time Step 196
7.4.2 Local Error Bound 197
7.4.3 Stability 199
7.4.4 Proof of Theorem 7.1 For-1¡Ü¦Á¡Ü0 200
7.5 Higher-Order Error Bounds In Lower-Order Sobolev Spaces 201
7.6 Numerical Experiments 204
7.7 Concluding Remarks 207
References 207
Chapter 8 Linearly-Fitted Conservative (Dissipative) Schemes For Nonlinear Wave Equations 210
8.1 Introduction 210
8.2 Preliminaries 212
8.3 Extended Discrete Gradient Method 215
8.4 Numerical Experiments 221
8.4.1 Implementation Issues 222
8.4.2 Conservative Wave Equations 223
8.4.3 Dissipative Wave Equations 230
8.5 Conclusions 232
References 233
Chapter 9 Energy-Preserving Schemes For High-Dimensional Nonlinear Kg Equations 235
9.1 Introduction 235
9.2 Formulation of Energy-Preserving Schemes 238
9.3 Error Analysis 243
9.4 Analysis of The Nonlinear Stability 245
9.5 Convergence 248
9.6 Implementation Issues of Kgdg Scheme 251
9.7 Numerical Experiments 255
9.7.1 One-Dimensional Problems 255
9.7.2 Two-Dimensional Problems 260
9